Maria Laura Delle Monache 1

Transcription

1 ESAIM: PROCEEDINGS AND SURVEYS, September 24, Vol. 45, p J.-S. Dhersin, Editor MODELING OF MOVING BOTTLENECKS IN TRAFFIC FLOW: A PDE-ODE MODEL WITH MOVING DENSITY CONSTRAINTS Maria Laura Delle Monache Abstract. In this article we study a strongly coupled PDE-ODE problem from an analytical and numerical point of view. This model describes a moving bottleneck that can be produced by the presence of a bus on a road network. The PDE is a scalar hyperbolic conservation law describing the traffic flow while the ODE, that describes the bus traectory, is intended in Carathéodory sense. The moving constraint is given by an inequality on the flux which accounts for the bottleneck created by the bus on the road. We give an existence result and introduce a finite volume scheme. The algorithm uses a locally non-uniform moving mesh which tracks the bus position. Some numerical tests are presented, in order to describe the behavior of the solution. Résumé. Dans cet article nous étudions un modèle couplé EDP-EDO d un point de vue analytique et numérique et nous vérifions le caractère bien posé de ce problème. Le modèle a été introduit pour décrire un goulot d étranglement mobile. En particulier, il peut décrire le mouvement d un bus, qui roule à une vitesse inférieure à celle des autres voitures, réduisant la capacité de la route et générant un rétrécissement mobile de la chausée. L EDP est une loi de conservation scalaire avec une contrainte mobile sur la densité et l EDO décrive la traectoire du bus. Nous présentons un résultat d existence des solutions du modèle, obtenu par la méthode d approximation de suivi des fronts, et nous montrons des simulations numériques obtenues avec une méthode front-capturing. Introduction In the 95s Lighthill and Whitham [7] and, independently, Richards [8] proposed the first macroscopic model for traffic known as the LWR model. They were the first ones to consider traffic flow as a fluid and to use fluid dynamics to describe its behavior on a single road. The traffic evolution is described with a non-linear scalar hyperbolic partial differential equation (PDE). In the last decade, several authors proposed different models to track the movement of a single car or a single vehicle among traffic, see [2,5,8,6] and references therein. In all these works the dynamics of the single vehicle is described by an ordinary differential equation (ODE). In this article, we focus on a strongly coupled PDE-ODE problem that was introduced in [3] to describe the effects of urban transport systems in a road network. The main traffic is described with a non-linear transport equation while the bus traectory is described with an ODE and with an inequality constraint which describes the drop in the capacity of the road due to the presence of the slower vehicle, for example a bus. The solution of the ODE will be intended in Carathéodory sense. Compared to the previous models, the present one gives a more realistic description of the velocity of the slower vehicle and it is easier to handle both from the analytical This research was supported by the European Research Council under the European Union s Seventh Framework Program (FP/27-23) / ERC Grant Agreement n Inria Sophia Antipolis-Méditerranée; maria-laura.delle monache@inria.fr c EDP Sciences, SMAI 24 Article published online by EDP Sciences and available at or

2 ESAIM: PROCEEDINGS AND SURVEYS 457 and the numerical point of view. This model can be considered a generalization of the model with fixed constraint studied in [,6,7] to model toll gates and pedestrian flow. Subsequently, some numerical methods have been developed, see [9,, 2]. In this article, we give an existence result for the Cauchy problem with BV data and provide a finite volume algorithm which tracks the bus at each time step using a non-uniform moving mesh. The article is developed as follows. Section gives a description of the model from an analytical point of view. Section 2 is dedicated to the solution of the Riemann problem and Section 3 shows the existence of solutions for the Cauchy problem. Section 4 and 5 are dedicated to the numerical scheme and show some numerical tests.. Mathematical model The goal of this work is to describe the phenomena caused by the presence of a bus in a car flow. Since the macroscopic description of the traffic does not allow to consider single vehicles, we consider the bus as a mobile obstacle that reduces the capacity of the road generating a moving bottleneck for the surrounding traffic. The traffic evolution is described by a scalar hyperbolic conservation law t ρ+ x f(ρ) =, () where the main quantities are the mean traffic density ρ = ρ(t,x) [,R] which is the scalar conserved quantity, with R being the maximal density allowed on the road and the flux function f : [,R] R + which is a strictly concave function such that f() = f(r) =, see Figure a. It is given by the following flux-density relation f(ρ) = ρv(ρ), where v is a smooth decreasing function denoting the mean traffic speed and here set to be v(ρ) = V( ρ R ), V being the maximal velocity allowed on the road. f(ρ) f(ρ) Two lanes f max One lane ρ cr R ρ ρα,cr ρcr α R ρ (a) One lane (b) Several lanes Figure. Fundamental Diagram Every road has a specific fundamental diagram. For example, for roads with different number of lanes we can consider as a first approximation that the fundamental diagrams are not affine and in a ratio that depends on the maximal capacities of the different lanes, see Figure b. The bus does not behave like cars hence, it cannot be modeled in the same way. We represent a single bus such that we can track its traectory at all times. When it is possible, the bus will move at its own maximal speed which, we denote as V b < V. When the surrounding traffic is too dense the bus will adapt its velocity to the one of the cars, so it will not be possible for the bus to overtake the cars. From a mathematical point of view, the velocity of the bus can be described by the following function: Vb if ρ ρ ω(ρ) =. = R( V b V ), (2) v(ρ) otherwise,

3 458 ESAIM: PROCEEDINGS AND SURVEYS and the bus traectory is described by the following ODE ẏ(t) = ω(ρ(t, y(t)+)) (3) where y denotes the position of the bus. To describe the interaction between the bus and the traffic we consider the bus as a mobile obstacle, i.e., as a moving restriction of the road. The situation is the following: upstream and downstream with respect to the bus, the cars behave normally while on the side of the bus the road capacity is reduced, generating a bottleneck, see Figure 2. V b V b Bus V b Figure 2. Moving bottleneck This discontinuity moves at the bus speed. To better capture the influence of the bus, we choose to study the problem in the bus reference frame. This means setting X = x y(t). In this coordinate system the velocity of the bus is equal to zero. As a consequence the conservation law can be rewritten as: t ρ+ X (f(ρ) ẏρ) =, (4) with a corresponded a constraint on the flux that can be written in the following way f(ρ(t,y(t))) ẏ(t)ρ(t,y(t)) αr 4V (V ẏ(t))2 (5) with the constant coefficient α ],[ giving the reduced rate of the road capacity due to the presence of the bus. This constraint is obtained setting f(ρ(t,y(t))) ẏρ(t,y(t)) f α (ρ α ) ẏρ α (6) where f α (ρ) = ρ ( α) ρ is the smaller fundamental diagram in Figure 3 and ρα = α 2 ( ẏ) the point that satisfies f α(ρ α ) = V b. This constraint is always satisfied if ẏ(t) = v(ρ) since the left hand side is. Moreover, it is well defined because of the Rankine-Hugoniot conditions. This constraint describes mathematically the reduction of capacity of the road due to the presence of the bus. For our purposes, it is not restrictive to assume that R = V =, so that the full model writes t ρ+ x (ρ( ρ)) =, (t,x) R + R, ρ(,x) = ρ (x), x R,. α f(ρ(t,y(t))) ẏ(t)ρ(t,y(t)) F α = 4 ( ẏ(t))2, t R +, ẏ(t) = ω(ρ(t,y(t)+)), t R +, y() = y. The above model was introduced in [3], in an engineering framework, to study the effect of urban transport systems in a road network. Its analytical properties were addressed in []. Here we show some results but we refer the reader to [] for a detailed explanation and proofs. (7)

4 ESAIM: PROCEEDINGS AND SURVEYS The Riemann problem with moving density constraint Consider (7) with the particular choice of initital data y = and ρ (x) = ρl if x <, ρ R if x >. (8) We are now ready to define the Riemann solver for (7), (8) following [3, V]. First, we recall the definition for a classical Riemann problem Definition 2.. The classical Riemann solver R for (),with initial datum ρ (x) = ρl if x <, ρ R if x >. (9) is the (right continuous) map (t,x) R(ρ L,ρ R )( x t ) that gives the standard weak entropy solution to (). It is defined as follows. If ρ L < ρ R, then the entropy-admissible solution is given by the shock wave ρl if x < λt, ρ(t,x) = if x > λt. ρ R () where, by the Rankine-Hugoniot condition we get λ = f(ρ R) f(ρ L ). If, instead, ρ L > ρ R the entropyadmissible solution to the Riemann problem is given by the rarefaction wave ρ R ρ L ρ L if x < f (ρ L )t, ρ(t,x) = (f ) ( x t ) if f (ρ L )t < x < f (ρ R )t, () ρ R if x > f (ρ R )t. f(ρ) f(ρ) V b F α ˇρ α ρ α ˆρ α α ρ ˇρ α ρ α ˆρ α ρ ρ max V b ρ (a) Fixed reference frame (b) Bus reference frame for ẏ = V b Figure 3. Flux functions. The big fundamental diagram describe the whole road and the smaller one the piece of road on the side of the bus. Let ˇρ α and ˆρ α, with ˇρ α ˆρ α, be the two values of the density among which the flux constraint acts (see Figure 3) then, we can define the following.

5 46 ESAIM: PROCEEDINGS AND SURVEYS Definition 2.2. The constrained Riemann solver R α for (7), (8) is defined as follows. () If f(r(ρ L,ρ R )(V b )) > F α +V b R(ρ L,ρ R )(V b ), then R α (ρ L,ρ R )(x) = R(ρL, ˆρ α ) if x < V b t, R(ˇρ α,ρ R ) if x V b t, and y(t) = V b t. (2) If V b R(ρ L,ρ R )(V b ) f(r(ρ L,ρ R )(V b )) F α +V b R(ρ L,ρ R )(V b ), then (3) If f(r(ρ L,ρ R )(V b )) < V b R(ρ L,ρ R )(V b ), then R α (ρ L,ρ R ) = R(ρ L,ρ R ) and y(t) = V b t. R α (ρ L,ρ R ) = R(ρ L,ρ R ) and y(t) = v(ρ R )t. Note that, when the constraint is enforced (point () in the Definition 2.2), a non-classical shock arises, which satisfies the Rankine-Hugoniot condition but violates the Lax entropy condition. 3. The Cauchy problem: existence of solutions The aim of this section is to study the existence of the solutions of problem (7). We will prove existence of solutions of both the conservation law with the constraint and the ODE, at the same time. As seen before, the bus influences the traffic along the road but it is also influenced by it. The bus position y = y(t) solves ẏ(t) = ω(ρ(t,y(t)+)), (2) y() = y. Solutions to (2) are to be intended in Carathéodory sense, i.e., as absolutely continuous functions which satisfy (2) for a.e. t. We give our definition of solution. Definition 3.. A couple (ρ,y) C ( R + ;L BV(R;[,R]) ) W, (R + ;R) is a solution to (7) if () ρ is a weak solution of the conservation law, i.e., for all ϕ C c(r 2 ;R) (ρ t ϕ+f(ρ) x ϕ)dx dt+ ρ (x)ϕ(,x) dx = ; R + R R moreover, ρ satisfies the Kružhkov entropy conditions [5] on (R + R) \ (t,y(t)): t R + }, i.e., for every k [,] and for all ϕ C c(r 2 ;R + ) and ϕ(t,y(t)) =, t >, ( ρ k t ϕ+sgn(ρ k)(f(ρ) f(k)) x ϕ)dx dt+ ρ k ϕ(,x) dx ; (3b) R + R R (2) y is a Carathéodory solution of the ODE, i.e., for a.e. t R + (3a) y(t) = y + t ω(ρ(s, y(s)+)) ds ; (3c) (3) the constraint is satisfied, in the sense that for a.e. t R + lim (f(ρ) ẏρ)(t,x) F α. x y(t)± (3d) Remark that the above traces exist because ρ(t, ) BV(R) for all t R +. The following theorem ensures the existence of solutions.

6 ESAIM: PROCEEDINGS AND SURVEYS 46 Theorem 3.2. Let ρ BV(R;[,R]), then problem (7) admits a solution in the sense of Definition 3.. In the rest of the section we give a sketch of the proof of Theorem 3.2. In particular, we will explain how to construct a sequence of approximate solutions via the wave-front tracking method, and prove its convergence. Finally, we will check that the limit functions satisfy conditions (3a)-(3d) of Definition 3.. For the detailed proof we address the reader to []. 3.. Wave-front tracking To construct piecewise constant approximate solutions, we adapt the standard wave-front tracking method, see for example [3, 6]. Fix a positive n N, n > and introduce in [,] the mesh M n defined by M n = (2 n N [,]) ˇρ α, ˆρ α },modifiedinordertoincludethecriticalpoints ˇρ α, ˆρ α. Moreover, letf n bethepiecewise linear function thatcoincides with f onm n, andletρ n = ρ,χ [x,x [ beapiecewiseconstant function with values in M n which approximates ρ in the sense of the strong L topology and such that TV(ρ n ) TV(ρ ). We set x = y. For small times t >, a piecewise approximate solution (ρ n,y n ) to (7) is constructed piecing together the solutions to the Riemann problems t ρ+ x (f n (ρ)) =, ρ(,x) = ρ, if x < y, ρ, if x > y, f(ρ(t,)) ẏ n ρ(t,) α 4 ( ẏ n) 2 t ρ+ x (f n (ρ)) =, ρ(,x) =, ρ, if x < x, ρ,+ if x > x, where y n satisfies ẏn (t) = ω(ρ n (t,y n (t)+)), (5) y n () = y. The approximate solution ρ n constructed above can be prolonged up to the first time t >, where two waves collide, or a discontinuity hits the bus traectory. In both cases, a new Riemann problem arises and its solution, obtained in the former case with R and in the latter case with the constrained Riemann solver R α, allows to extend ρ n further in time Bounds on the total variation Due to the presence of non-classical shocks, the total variation may increase at interactions. To provide an uniform bound, we introduce the following Glimm-type functional. Given an approximate solution ρ n = ρ n (t, ) constructed by the wave-front tracking method, we define where γ is given by Υ(t) = Υ(ρ n (t, )) = TV(ρ n )+γ = (4) ρ n + ρ n +γ, (6) γ = γ(t) = if ρ n (t,y n (t) ) = ˆρ α, ρ n (t,y n (t)+) = ˇρ α 2 ˆρ α ˇρ α otherwise. (7) The value of γ is chosen to have the following uniform bound on Υ. Lemma 3.3. For any n N, the map t Υ(t) = Υ(ρ n (t, )) at any interaction either decreases by at least 2 n, or remains constant and the number of waves does not increase. Lemma 3.3 in particular implies that the total number of waves is finite and therefore the wave-front tracking procedure can be prolonged to any time T >. Moreover, it holds TV(ρ n (t)) γ(t) γ() = TV(ρ n ) + 2 ˆρ α ˇρ α. For the proof we refer the reader to [].

7 462 ESAIM: PROCEEDINGS AND SURVEYS 3.3. Convergence of approximate solutions In this section we prove that the limit of wave-front tracking approximations provides a solution (ρ, y) of the PDE-ODE model (7) in the sense of Definition 3.. The following Lemma states the convergence of the wave-front tracking approximations. Lemma 3.4. Let ρ n and y n, n N, be the wave-front tracking approximations to (7) constructed as detailed in Section 3., and assume TV(ρ ) C be bounded, ρ. Then, up to a subsequence, we have the following convergences ρ n ρ in L loc(r + R); (8a) y n ( ) y( ) in L ([,T]), for all T > ; (8b) ẏ n ( ) ẏ( ) in L ([,T]), for all T > ; (8c) for some ρ C ( R + ;L BV(R) ) and y W, (R + ). Proof. (8a) is recovered using a standard procedure based on Helly s Theorem (see [3, Theorem 2.4]). (8b) is given by an application of the Ascoli-Arzelà Theorem. In order to prove (8c), we show that TV(ẏ n ;[,T]) is uniformly bounded. For further details on this proof, we refer the reader to []. Proof of Theorem 3.2. Since ρ n converge strongly to ρ in L loc (R+ R), therefore it is straightforward to pass to the limit in the weak formulation of the conservation law, proving that the limit function ρ satisfies (3a). Kružhkov entropy condition (3b) can be recovered in the same way. To prove 3c we need to show that lim n ρn (t,y n (t)+) = ρ + (t) = ρ(t,y(t)+) for a.e. t R +. (9) holds. At this purpose, we use the weak convergence of measures in [4, Lemma 5] and we proceed like in [4, 4], for the details see []. Combining (8c) and (9) we get ẏ(t) = ω(ρ(t,y(t)+)) for a.e. t >. In order to verify that the limit solutions satisfy the constraint (3d), we can use directly the convergence result proved in the previous section. 4. Numerical scheme In this section we present a numerical method to compute solutions, including non-classical ones, to strongly coupled constrained PDE-ODE problems. We apply a front tracking capturing method which uses a Lagrangian algorithm in which the interface is tracked, such as the one in [9], together with a numerical method that tracks at each time step the slower vehicle traectory, as in [5]. 4.. Godunov-type scheme for hyperbolic PDEs with constraint We use the following notation: x n are the cell interfaces at time t n with n N and Z. A computational + 2 cell is given by [x n,x n ] where x n 2 + is the center of the cell and hn = xn x n is the cell width at time t n. Since we know the exact solutions of the Riemann Problem we develop a Godunov-type scheme. Classical Godunov scheme [4] can be expressed in conservative form as U n+ = U n t ( ) F(U n,u n x +) F(U,U n n ), (2) where F(U,V) is the numerical flux. Boundary conditions are imposed on the left and on the right ends of the computational domain. Classical Godunov scheme does not represent all the characteristics of the solutions,

8 ESAIM: PROCEEDINGS AND SURVEYS 463 failing to represent non-classical shocks. Since our aim is to track the traectory of the bus, and, also, represent non-classical solutions, a Lagrangian algorithm with a moving mesh is used. In particular, we develop an algorithm which follows at each time step the bus traectory and modifies the mesh when the constraint enforces, that is when f(r(ρ L,ρ R )(V b )) > F α +V b R(ρ L,ρ R )(V b ). (2) To keep non-classical shock on cell interfaces, we shift grid points locally and, as a consequence, we have a locally non-uniform mesh due to a cell interface moving with the bus traectory. We use the superscript new to indicate the quantities that are modified at time t n with the grid. Assume that at time t n, y n is the bus position and y n ]x n,x n ] for some m. When the constraint (2) is enforced, the algorithm for the adaptive mesh m 2 m+ 2 reads as follows: If x n y n h n m+ > m 2 2 then move the point xn m 2 cells m and m from the formula to x new = y n and recompute the cell averages in the m 2 x n m U Um new m n +(x new x n m = 2 m 2)U n m x new m (22) with x new m = x new m 2 x n y n h n m+ m 2 x n. m 3 2 If 2 we adust the location of the point xn m+ 2 place the point x new at the middle distance between x n m 2 m 3 2 averages in the cells m and m+ from the formulas and x new m+ 2 such that x new m+ 2 = y n and then we. We then compute the new cell U new m = U new m+ = (x n x new )U m 2 m m n +(x new x n 2 m+ 2 m+ 2)U n m x new m x n Um+ n +(x n x new m+ 2 m+ 2)U n m x new m+ (23) (24) with x new m = x new x new m+ 2 m 2 and x new m+ = x new x n. m 2 m 3 2 Each time the constraint is enforced the bus position follows the non-classical shock traectory: y n+ = x n+ = m± 2 x new +V m± b t n. The other cell interfaces are kept unchanged. 2 An explicit formula for the scheme can be derived and looks as follows U n+ = xn x n+ U n xn [( t n+ ) x n+ f(r(u n,u+)) V n b R(U n,u+) n dt t n ) ] ( t n+ f(r(u,u n n )) V b R(U,U n n ) dt. (25) t n For simplicity we introduce the notation F(U) = F(U) V b U. Notice that in our case F corresponds to the F α when the constraint is active and to F(U) with V b = when the constraint is not enforced. Moreover, one needs to be careful when the bus traectory is dealt with. In fact, the cell size changes as time increases. In fact, after the mesh has been resized and adusted, we update the cell averages for all cells with the following conservative formula: U n+ = xnew x n+ U n tn n ( F(U x n+,u+) n F(U,U n )) n. (26)

9 464 ESAIM: PROCEEDINGS AND SURVEYS.9 Mean traffic density Bus traectory.9 Mean traffic density Bus traectory Time= Time= Mean traffic density Bus traectory.9 Mean traffic density Bus traectory Time= Time=.769 Figure 4. Evolution of the density at different times corresponding to initial data (27) and a mesh grid of 5 points Numerical method for the ODE Ateachtimet n wedeterminethepositiony n ofthedriverbystudyinginteractions betweenthebustraectory and the density waves within a fixed cell. We distinguish two cases: Constraint (2) is satisfied. Then the bus moves always at velocity V b and the bus position is updated y n+ = V b t n +y n. Constraint (2) is not satisfied. In this case we implement the tracking algorithm introduced in [5]. 5. Numerical results In this section we present a numerical test performed with the scheme previously described. We display some numerical results obtained applying the modified Godunov scheme to problem (7). Here we deal with a road of length parameterized by the interval [,]. In all the simulations we fix V b =.3, α =.6. We consider the following initial data.8 if x <.5, ρ(,x) = y.53 if x >.5, =.5. (27) The values of the initial conditions create a rarefaction wave followed by a non-classical and a classical shocks on the density, as illustrated in Figure 4 and 5. Note the non-classical downward ump at the bus location. 6. Conclusions This article introduces a strongly coupled PDE-ODE problem with a moving density constraint which can model the presence of a moving bottleneck on a road network. We proved the existence of solutions for general BV data for the Cauchy problem and proposed a Lagrangian finite volume scheme that is able to display non-classical solutions. Stability of the problem and a conservative numerical method are under study.

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